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Say, hypothetically, you have two terrestrial planets (b,c), one gas dwarf(d), and one gas giant(e) all orbit the habitable zone in some manner.

b, c, and d all have an orbital resonance, which cause an eccentricity of 0.04. They are always in the habitable zone.

e has a very high eccentricity of 0.93, causing it to go on the outer fringe of the habitable zone, then go out into the frigid outer solar system. While e passes very near d, it does not orbit its star on the same plane as d.

Let's pretend that this star is an M-type star, around 0.45 solar masses.

I want to know if this is theoretically possible.

Edit: Since it seems impossible permanently, I decided to put the gas giant far away from the habitable zone so the planets could be stable. Thanks for the help!

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    $\begingroup$ Another possibility is to have b and c be moons of d. If so, make sure you take irradiance from d itself into consideration when calculating the temperature of b & c (you'll want d to be farther out from the star than normal, as d will warm b & c somewhat. Exact amount depends on how large d is.) $\endgroup$
    – TLW
    Commented Sep 29 at 19:57

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I'm sorry to say that it is not possible to have these four planets independantly orbiting an M-type star of 0.45 solar masses in its habitable zone.

Why? Because the habitable zone of an M-type star of this size would be rather small, and if these worlds were orbiting that close to each-other, the probability of the gas giant's gravitic field interacting with the other worlds and causing them to be ejected, captured or swallowed would approach a certainty within a very short amount of time on an astronomical time-scale.

However, all is not lost... there are other ways.

Obviously the gas giant is going to have the most gravitic influence over this system. Like jupiter in our own solar system, it'll sweep up all sorts of debris from the solar system. However, the gas dwarf could well orbit it (or a common barycentre), and if two satellites of this gas giant/dwarf binary were big enough to be terrestrial planets, they could all orbit their M-type star in its habitable zone long enough for these earth-sized worlds to generate their own life... or have life transplanted from a nearby world.

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    $\begingroup$ of course another rather serious problem would be the extreme radiation environment. These moons and planetoids would be constantly bathed in a thick soup of X-rays and other not so healthy stuff, making survival of life, as we know it, impossible. $\endgroup$
    – jwenting
    Commented Sep 30 at 13:03
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Additional point of meh:

While e passes very near d, it does not orbit its star on the same plane as d.

Due to mutual interaction and momentum exchange, planets will end up orbiting all in a plane well before they are formed. That's what we call ecliptic for our solar system.

The only way for what you describe to happen is that either:

  • planet e is not actually a planet but rather a minor body
  • planet e has been recently captured by the system

Note that the second possibility means that your system is going to change quite drastically in the near future due to the chaotic interaction of the new arrival with the rest of the planets.

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    $\begingroup$ The planets orbit the Sun only roughly in the same plane. The ecliptic is specifically the plane of Earth's orbit. Mercury's orbit is inclined 7° with respect to the ecliptic, Venus's 3.4°, Saturn's 2.5°, and so on. That's why the Zodiac is not a line, but rather a band about 16° wide. $\endgroup$
    – AlexP
    Commented Sep 29 at 8:27

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